The natural operators lifting 1-forms on manifolds to the bundles of \(A\)- velocities. (English) Zbl 0823.58004

Mikulski, W. M.

Monatsh. Math. 119, No. 1-2, 63-77 (1995).

Let \(A\) denote a Weil algebra on \(p\) variables. In this paper it is shown that for any \(n\)-manifold with \(n\geq p+ 2\) the set of all natural operators \(T^*\to T^* T^ A\) is a free finitely generated module over a ring canonically dependent on \(A\).

Reviewer: A.Dimca (Bordeaux)

MSC:

58A20

Jets in global analysis

Keywords:

fibre bundles; Weil algebra; \(n\)-manifold

References:

[1]	Gancarzewicz, J., Mahi, S.: Lifts of 1-forms to the tangent bundle of higher order. Czech. Math. J.40(115), 397-407 (1990). · Zbl 0729.58003
[2]	Gancarzewicz, J., Mikulski, W. M., Pogoda, Z.: Lifts of some tensor fields and connections to product preserving functor. Nagoya Math. J. (to appear). · Zbl 0813.53010
[3]	Kol??, I., Michor, P. W., Slov?k, J.: Natural Operations in Differential Geometry. Berlin: Springer. 1993. · Zbl 0782.53013
[4]	Kurek, J.: On the first order natural operators transforming 1-forms on a manifold to linear frame bundle. Demonstratio Math.26, 287-293 (1993). · Zbl 0791.53018
[5]	Kurek, J.: On the first order natural operators transforming 1-forms on a manifold to the tangent bundle. Ann. U.M.C.S.43, 79-83 (1989). · Zbl 0739.58001
[6]	Mikulski, W. M.: Some natural operations on vector fields. Rend. Math. (Serie VII)12, 783-803 (1992). · Zbl 0766.58005
[7]	Mikulski, W. M.: Some natural constructions on vector fields and higher order cotangent bundles. Mh. Math.117, 107-119 (1994). · Zbl 0796.58002 · doi:10.1007/BF01299315
[8]	Mikulski, W. M.: Natural transformations transforming functions and vector fields to functions on some natural bundles. Math. Bohemica117, 217-223 (1992). · Zbl 0810.58004
[9]	Mikulski, W. M.: Natural transformations transforming vector fields into affinors on the extendedr-th order tangent bundles. Arch. Math. Brno29, 59-70 (1993). · Zbl 0804.58005
[10]	Mikulski, W. M.: Natural liftings of foliations to the tangent bundle. Math. Bohemica117, 409-414 (1992). · Zbl 0774.53019
[11]	Morimoto, A.: Prolongations of connections to bundles of infinitely near points. J. Diff. Geom.11, 479-498 (1976). · Zbl 0358.53013
[12]	Morimoto, A.: Liftings of tensor fields and connections to tangential fibre bundles of higher order. Nagoya Math. J.40, 85-97 (1970). · Zbl 0208.50102
[13]	Weil, A.: Theorie des points proches sur les varietes differentiables. In: G?om?trie Differentielle. Colloque du C.N.R.S. Strasbourg, pp. 111-117, 1953.
[14]	Yano, K.;Ishihara, S.: Differential geometry in tangent bundles. Kodai Math. Sem. Rep.18, 271-292 (1966). · Zbl 0146.17702 · doi:10.2996/kmj/1138845273